Hi Marc,
avast! Mobile Security doesn't interfere with Play Store, it just needs it to be present on the device. From what I see here:
http://support.google.com/googleplay/bin/answer.py?hl=en&answer=1727131, your device isn't on the list of supported device. This means that the device isn't licensed to have some Google apps installed, like Play store, Gmail etc.
If you haven't installed Play store yourself (like it's possible with some custom ROMs), you probably don't have it on the device. I'm not sure how it could disappear if it was there already.
What is meant by the "it must have at least one logged in Google account" is, that what you go to Settings - Accounts, there must be a Google account present. This account is then used in Play Store, Gmail, Calendar, Contacts etc.
I'm afraid I can't help you more, because we don't have this device here. I'm just guessing from what I can find on the internet.
Ondra